Reduced groups with action were introduced in [1]} as a certain full subcategory \(\mathbf {rGr}^{\bullet }\)  of the category of groups with action, which was defined in [2]}, [3]} and played a crucial role in the solution of Loday's two problems stated in [4]}, [5]}. The aim of the definition of \(\mathbf {rGr}^{\bullet }\)  was investigation of representability of actions in this category. The notion of representable action was introduced in [6]} in terms of monoidal categories and applied to semi-abelian categories. In [7]}, [8]} was developed action theory in categories of interest in the sense of [9]}, [10]}, where was given a definition and construction of universal strict general actor \(\mathsf {USGA}(A)\)  of any object \(A\)  in the category of interest and found its properties. This construction and the corresponding results were applied to various cases of algebraic categories in order to prove the existence of universal acting objects, called actor, under appropriate conditions and to give their constructions [11]}, [7]}, [8]}, [14]}, [15]}, [16]}, [17]}, [18]}. The notion of actor is equivalent to split extension classifier in semi-abelian categories [6]}. The category \(\mathbf {Gr}^{\bullet }\)  is not a category of interest, therefore we found its subcategory \(\mathbf {rGr}^{\bullet }\) , which is also not a category of interest, but it has some properties, which are similar to those which are important in the construction of \(\mathsf {USGA}(A)\)  of an object \(A\) . In this paper we apply the results obtained in [8]} and [1]}, and introduce a notion of pentactions of \(A\)  in \(\mathbf {rGr}^{\bullet }\) . It is proved that if an object \(A\)  is perfect with zero weak stabilizer in the sense defined in the paper, then pentactions of \(A\)  form an object of \(\mathbf {rGr}^{\bullet }\) , denoted by \(\mathsf {Pentact}(A)\) , \(A\)  is action representable and \(\mathsf {Pentact}(A)\)  represents all actions on \(A\) .
